[Merged by Bors] - refactor(group_theory/order_of_element): now makes sense for infinite monoids #6587
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Co-authored-by: Eric Wieser <wieser.eric@gmail.com>
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… monoids (#6587) This PR generalises `order_of` from finite groups to (potentially infinite) monoids. By convention, the value of `order_of` for an element of infinite order is `0`. This is non-standard for the order of an element, but agrees with the convention that the characteristic of a field is `0` if `1` has infinite additive order. It also enables to remove the assumption `0<n` for some lemmas about orders of elements of the dihedral group, which by convention is also the infinite dihedral group for `n=0`. The whole file has been restructured to take into account that `order_of` now makes sense for monoids. There is still an open issue about adding [to_additive], but this should be done in a seperate PR. Also, some results could be generalised with assumption `0 < order_of a` instead of finiteness of the whole group. Co-authored-by: Julian-Kuelshammer <68201724+Julian-Kuelshammer@users.noreply.github.com> Co-authored-by: Eric Wieser <wieser.eric@gmail.com>
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… monoids (#6587) This PR generalises `order_of` from finite groups to (potentially infinite) monoids. By convention, the value of `order_of` for an element of infinite order is `0`. This is non-standard for the order of an element, but agrees with the convention that the characteristic of a field is `0` if `1` has infinite additive order. It also enables to remove the assumption `0<n` for some lemmas about orders of elements of the dihedral group, which by convention is also the infinite dihedral group for `n=0`. The whole file has been restructured to take into account that `order_of` now makes sense for monoids. There is still an open issue about adding [to_additive], but this should be done in a seperate PR. Also, some results could be generalised with assumption `0 < order_of a` instead of finiteness of the whole group. Co-authored-by: Julian-Kuelshammer <68201724+Julian-Kuelshammer@users.noreply.github.com> Co-authored-by: Eric Wieser <wieser.eric@gmail.com>
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This PR generalises
order_offrom finite groups to (potentially infinite) monoids. By convention, the value oforder_offor an element of infinite order is0. This is non-standard for the order of an element, but agrees with the convention that the characteristic of a field is0if1has infinite additive order. It also enables to remove the assumption0<nfor some lemmas about orders of elements of the dihedral group, which by convention is also the infinite dihedral group forn=0.The whole file has been restructured to take into account that
order_ofnow makes sense for monoids. There is still an open issue about adding [to_additive], but this should be done in a seperate PR. Also, some results could be generalised with assumption0 < order_of ainstead of finiteness of the whole group.